Pythagoras’ theorem shows that the sum of the squares of the two shorter sides of a right-angled triangle is equal to the square of its longest side (the hypotenuse).

A hypotenuse is the side of the right-angled triangle which is opposite the right-angle.

A right-angled triangle has exactly one 90° interior angle.

Length of the hypotenuse

Maths

Checking understanding of similarity

Checking understanding of congruence

Similarity in shapes

Congruence in shapes

Congruent, similar or neither

Rotational symmetry

Congruent triangles (SSS)

Congruent triangles (SAS)

Congruent triangles (ASA and AAS)

Congruent triangles (RHS)

Applying the criteria for congruence

Demonstrating Pythagoras' theorem

Further demonstrating of Pythagoras' theorem

Length of a shorter side

Determining which side

Pythagoras' theorem in context

Problem solving with similarity and Pythagoras' theorem

In this unit, pupils continued to build on their understanding of congruence and similarity with right-angled triangles. In the future unit, similarity is used to extend a right-angled triangle in the unit circle to any other right-angled triangle.

In the prior unit, pupils explored the effects of different transformations and whether the transformed shape was congruent or similar to the original shape. This is built on in this unit where pupils formalise the criteria for determining congruence in triangles and study rotational symmetry.

This unit explores the concepts of congruence and similarity, and ways to prove or disprove similarity between shapes. We then explore further properties of triangles with Pythagoras' Theorem.

Geometrical properties: similarity and Pythagoras' theorem

The length of the hypotenuse for this right-angled triangle is {{ }} cm.

$$k$$ cm² is the area of the square from the hypotenuse of the triangle. The value of $$k$$ is {{}}.

Calculate the length of the hypotenuse of this triangle, in units.

The area, $$w$$, of the largest square in this diagram is {{}} cm².

Calculate the length of the hypotenuse for this triangle. (Give your answer to 1 d.p.).

The length of the hypotenuse of this triangle, rounded to 1 d.p. is {{}} cm.

The triangle formed from these three squares is right-angled. What is the value of $$n$$, where $$n$$ units² is the area of a square.

If three squares with different areas are joined at their vertices, what type of triangle would be formed?

Angle $$a°$$ is an acute angle, but is the largest angle in this triangle. Which of these are possible values of $$t$$, where $$t$$ units² is the area of a square.

Describes a property of only equilateral triangles.

Describes a property of only right-angled triangles.

Describes a relationship between the squares of the side lengths of a triangle.

Describes a relationship between the interior angles of a triangle.

Describes a relationship between squares formed from the sides of a triangle.

Which of these are true for the Pythagoras' theorem?

A right-angled triangle is formed from three squares. The area of two of the squares are 75 units² and 25 units². What are the possible areas of the third square?

Length of the hypotenuse

Lesson outcome